Each set of data can be modeled by f(x)=a x^2+b x+c (a)...

Each set of data can be modeled by $f(x)=a x^{2}+b x+c$ (a) Write a linear system whose solution represents values of $a, b,$ and $c$ (b) Use technology to find the solution. (c) Graph $f$ and the data in the same viewing rectangle. (d) Make your own prediction using $f$. (Refer to the introduction to this section.) The table lists total iPod sales $y$ in millions $x$ years after 2004 $$ \begin{array}{cccc} x & 0 & 2 & 4 \\ y & 3 & 55 & 150 \end{array} $$

Each set of data can be modeled by $f(x)=a x^{2}+b x+c$
(a) Write a linear system whose solution represents values of $a, b,$ and $c$
(b) Use technology to find the solution.
(c) Graph $f$ and the data in the same viewing rectangle.
(d) Make your own prediction using $f$.
(Refer to the introduction to this section.) The table lists total iPod sales $y$ in millions
$x$ years after 2004
$$
\begin{array}{cccc}
x & 0 & 2 & 4 \\
y & 3 & 55 & 150
\end{array}
$$

Key Concepts

Graphing and Visualization

Graphing the quadratic function alongside the data points provides a visual representation of how well the function models the data. The process involves plotting the curve defined by f(x) = ax² + bx + c and marking the data points on the same coordinate system to easily assess the fit and trends.

Predictive Modeling

Using the quadratic function to predict values involves substituting new x-values into the function to estimate corresponding y-values. This process, known as predictive modeling or regression, allows one to extrapolate trends from the existing data and forecast future outcomes.

Solving Systems of Equations

Once the system of linear equations is established, various techniques, including substitution, elimination, or technology-assisted methods like matrix operations or graphing calculators, can be used to find the unique values of the coefficients a, b, and c that best fit the data.

Linear Systems from Data Points

When a quadratic function is used to model data, plugging in the x and y values from given data points creates a system of linear equations in the coefficients a, b, and c. This method relies on the fact that a quadratic function is uniquely determined by three distinct points, allowing the setup of a solvable system of equations.

Quadratic Functions

A quadratic function is a polynomial of degree two, typically written in the form f(x) = ax² + bx + c. This function is used to model situations where the relationship between variables is non-linear, and its graph is a parabola, which can open upwards or downwards depending on the coefficient a.

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